Comportamiento del sonido dentro de un recinto

1. No-slip assumption - there is no slip between the wheels and the road. This allows us to relate the vehicle speed to the engine speed.

V = r*_{rw e}

2. Since platooning operations are high speed - high gear operation of the vehicle, we assume that there is negligible slip across the torque converter.

Note: With assumptions 1 and 2 we can reflect the wheel load, tload, to the engine side and rewrite the engine speed state equation as below.

&

( ) /

e

=

tnet

−

tload je where tload = r*(rwftrac+tbr).

Recalling the vehicle velocity state equation, we eliminate the tractive force, ftrac, between the V and ωe state equations to obtain an equation of the form below.

j e tnet r rw fdrag froll r tbr *

&

* *

( )

=

−

+

−

where j* = je + mass * r*2r2w

3. For vehicle position control applications the intake manifold dynamics are sufficiently faster, than engine dynamics. Hence we can write

&

&

ma1

≈

ma0

&

( / ) ( )

m_a0

≈

₁PRI p_m p TC_a

In view of the above equations we can simplify the net torque calculation to obtain tnet as a function of ωa and α.

The simplifying assumptions allow us to reduce the model to a simpler form for controller design. The model is represented by the following state equations.

&

* x

=

r _e

j*

&

_e

=

t_net

−

r r_w *(f_drag

+

_rollf )

−

r t* _br

&

( , ) /

Controller Design

Longitudinal vehicle control is achieved by one of 2 approaches - time headway or spacing distance control. The former involves control of the time of separation between succeeding vehicles while the latter is simply distance control. In this study we have adopted spacing control.

The controller design in the next few paragraphs deals with control of vehicles for platoon operations. The controller can be extended to take into account transition modes by replacing the constant desired spacing, spdes,i by a time varying desired spacing profile. The velocity regulation controller is slightly different but can be derived in a similar fashion.

A sliding mode approach has been followed. We define the spacing error e = xi - xi-1 - spdes,i

spdes,i is the desired spacing specified for platoon operation. We also define a scalar function, s1,

s1

= +

e c e1

+

c2

∫

edt

&

Differentiating the above function and expanding

e&&

we have

&

( ( ) )

&

&

&

* * , * * s r r j t r r f f r t x c e c e w

net des w drag roll br i

1

=

−

+

−

−

−1

+

1

+

2

In the above equation we can select tnet,des to achieve

&

s1

= −

k s1 1 for some positive k1. This ensures convergence of the errors to zero.

With knowledge of tnet,des and engine speed ωe we can calculate the desired throttle angle, des. In case αdes is less than a threshold we calculate the desired brake torque from the above equations. In order to determine the desired brake input tbr,c we have to define another scalar function, s2.

Differentiating this and in a procedure similar to the case for net desired torque we can calculate the desired brake torque such that

&

s₂

= −

k s_{2 2}.

Appendix C: Lateral Control Vehicle Models

The lateral vehicle model describes vehicle sprung mass dynamics in the six directions of motion, i.e., longitudinal(x), lateral(y), vertical(z), roll(φ), pitch(θ), and yaw(υ). The model does not include the engine dynamics but assumes that required torque at the wheels can be provided by the engine powerplant. This model was developed for a Toyota Celica by Peng. Vehicle Spring Mass Dynamics

Figure 52 shows the angles and dimensions associated with the vehicle in motion.

Figure 52. Vehicle Dimensions

Nomenclature

Mx,My,Mz : moments about x ,y,z direction

XYZ : inertial frame of reference.

Vx (Vy,Vz) : velocity in x (y,z) direction

FAi(FBi) : longitudinal (lateral) force on ith tire.

FPi : normal force on ith tire.

Froll : rolling resistance of tires.

Cdx, Cdy : drag coefficient in x,y direction. fwx, fwy : drag force in x,y direction.

m : mass.

h2 (h4,h5) : distances of c.g. from roll center (pitch center and ground).

Ix (Iy,Iz) : inertias about x y,z) axis

β : side slip angle

χ : velocity angle χ= ψ+β

The vehicle slip angle χis measured with respect to the sprung mass fixed co-ordinates while the vehicle yaw and velocity angles are measured with respect to the inertial coordinates. The relative yaw of the vehicle with respect to the road is given by ψc= ψψd. ψ is the actual vehicle yaw while ψd is the desired yaw as determined from the road curvature information.

The next few sections deal with the calculation of the tractive and suspension forces and moments required for deriving the equations of motion of the sprung mass dynamics.

Tire Forces

The tire forces are calculated from a tire model developed by Peng. The tire model was developed using the Bakker-Pacjecka-Lidner model. This is a model that has been obtained through experimental study followed by nonlinear curve fit to obtain the longitudinal and lateral tire forces fxi and fyi). Test data was obtained from a YOKOHAMA P205/60R1487H steel belted radial tire. The tire forces for the ith wheel are given by the functional expressions:

fxi = fx()i(i,fzi) fyi = fy()i(i,fzi)

where I, I are the slip ratio and slip angle respectively of the ith tire, and fzi is the normal force on the ith tire.

The slip ratio is defined as

i w wi x w wi

r

V

r

=

−

traction i w wi x x

r

V

V

=

−

braking

wi is the angular velocity of the ith wheel, and rw is the effective wheel radius.

The slip angle is the angle between the tire orientation plane and its velocity and is given by I = i - i

where ξi is the angle between the forward speed of the ith tire and the vehicle body, and δi is the wheel angle of the ith tire. We call this the steering angle.

tan(ζ1) ψ&_ψ& 1 1 2 = + − V l V s y x b tan( )

&

&

2 1 1 2

=

+

+

V l V s y x b tan( )

&

&

3 1 2 2

=

−

−

V l V s y x b tan( )

&

&

42 1 2 2

=

−

+

V l V s y x b

sb1, sb2 are the front and rear treadwidths respectively.

The above equations allow us to calculate tire forces under pure traction or cornering

maneuvers. Bakker proposed a method to correct these forces under the case of combined traction and cornering. For this we define normalized slip factors * and *

* max

=

* max

=

where, max and max are the values at which fxi and fyi achieve their peak values respectively. These are the fxi and fyi values from the traction only and cornering only tests. Now the correction factor is described as

*

_{=[( )}

* 2

_{+( ) ]}

* 2 5.

Then the longitudinal and lateral tire forces are corrected by the following two equations.

f

xi

=

f

x i

f

zi * * () *

(

,

)

f

yi

=

f

y i

f

zi * * () *

( ,

)

From these equations we can obtain the net longitudinal and lateral tire forces contributed by the ith wheel from the relations below.

FAi = fxicos( i) - fyisin( i) FBi = fxisin( i) - fyicos( i)

Suspension Forces

We have assumed a suspension system similar to what Peng used for his lateral control study. The suspension consists of a spring and damper at each wheel. The deflections of the

suspension joints can be calculated from geometry. The spring is assumed to be of the hardening type and is modeled as

In document Caracterización de la absorción sonora en modelos físicos a escala (página 79-91)